Approximations of Series Structural System Reliability: Method Comparison and Application to Timber Trusses

The exact failure probability of a series structural system requires multidimensional numerical integration that becomes computationally infeasible beyond ten to fifteen elements—a limitation routinely encountered in the probabilistic assessment of real structures such as timber trusses, bridge decks, and lattice girders. This paper presents a systematic formal analysis and comparison of the principal approximation methods available in the literature: simple bounds by Thoft-Christensen and Murotsu, Sørensen’s complement-product reformulation, and Ditlevsen’s narrow bounds based on pairwise failure event correlations. A new approximation is proposed, defined as the arithmetic mean of the upper and lower reliability bounds evaluated over the j most critical failure modes. Four formal propositions are established: (1) the approximation is bounded by the upper and lower bounds (consistency); (2) it is monotonically increasing in each component failure probability (monotonicity); (3) it converges to the exact midpoint as j → m (convergence); and (4) under positive inter-element correlation—the typical case for timber trusses manufactured from the same batch—the approximation overestimates the system reliability index below a crossover correlation ρ* (an unsafe-side error) and underestimates it above ρ* (a safe-side error), the practical timber truss regime falling on the unsafe side. Numerical validation against the exact solution for a ten-element series system across correlation levels ρ = 0 to 0.99, against five additional synthetic test systems, and against a classical equicorrelated benchmark from the reliability literature shows reliability-index errors of, at most, about 2.6%, anti-conservative below the crossover correlation and conservative above it. The approximation is applied to six timber truss configurations (spans 15–20 m, GL24h glulam, C24 structural timber, punched metal plate and bolt connections) within a probabilistic robustness assessment framework. Practical guidelines are provided for selecting the parameter j and for applying the approximation in robustness classification workflows.